That Higgs is trivial!

05/04/2015

ResearchBlogging.org

Notwithstanding LHC has seen the particle, the Higgs sector of the Standard Model has some serious problems. This fact yielded more than one headache to physicists. One of these difficulties is called technically “triviality“. Michael AizenmanThe scalar field theory, that is so well defined classically, does not exist as a quantum field theory unless is non-interacting. There is a wonderful paper by Michael Aizenman that shows that this is true for dimensions 5 and higher. So, one should think that, as we live in four dimensions, there is no reason to worry. The point is that Michael Aizenman left the question in four dimensions open. So, does Higgs particle exist or not and how does it yield mass if it will not interact? CERN said to us that Higgs particle is there and so, in some way, the scalar sector of the Standard Model must properly work. Aizenman’s proof was on 1981 but what is the situation now? An answer is in this article on Scholarpedia. As stated by the author Ulli Wolff

Triviality of lattice phi^4 theory in this sense has been rigorously proven for D>4 while for the most interesting borderline case D=4 we have only partial results but very strong evidence from numerical simulations.

While there is another great expert on quantum field theory, Franco Strocchi, in his really worth to read book saying

The recent proof of triviality of phi^4 in 3 + 1 spacetime dimensions indicates that the situation becomes worse in the real world, and in particular the renormalized perturbative series of the phi^4 model seems to have little to do with the non-perturbative solution.

We see that experts do not completely agree about the fact that a proof exists or not but, for sure, the scalar theory in four dimensions cannot interact and the Standard Model appears in serious troubles.

Before to enter more in details about this matter, let me say that, even if Strocchi makes no citation about where the proof is, he is the one being right. We have proof about this, the matter is now well understood and again we are waiting for the scientific community to wake up. Also, the Standard Model is surely secured and there is no serious risk about the recent discovery by CERN of the Higgs particle.

The proof has been completed recently by Renata Jora with this paper on arxiv. Renata extended the proof an all the energy range. I met her in Montpellier (France) at this workshop organized by Stephan Narison. We have converging interests in research. Renata’s work is based on a preceding proof, due to me and Igor Suslov, showing that, at large coupling, the four dimensional theory is indeed trivial. You can find the main results here and here. Combining these works together, we can conclude that Strocchi’s statement is correct but there is no harm for the Standard Model as we will discuss in a moment. Also the fact that the perturbation solution of the model is not properly describing the situation can be seen from the strictly non-analytical behaviours seen at strong coupling that makes impossible to extend what one gets at small coupling to that regime.

The fact that CERN has indeed seen the Higgs particle and that the Higgs sector of the Standard Model is behaving properly, unless a better understanding will emerge after the restart of the LHC, has been seen with the studies of the propagators of the Yang-Mills theory in the Landau gauge. The key paper is this where the behaviour of the running coupling of the theory was obtained on all the energy range from lattice computations.

Running Coupling

This behaviour shows that, while the theory is trivial at both the extremes of the energy range, there is an intermediate regime where we can trust the theory and treat it as an effective one. There the coupling does not run to zero but moves around some finite non-null value. Of course, all this is just saying that this theory must be superseded by an extended one going to higher energies (supersymmetry? Technicolor?) but it is reasonable to manage the theory as if all this just works at current energies. Indeed, LHC has shown that a Higgs particle is there.

So, triviality is saying that the LHC will find something new for sure. Today, beams moved again inside the accelerator. We are eager to see what will come out form this wonderful enterprise.

Aizenman, M. (1981). Proof of the Triviality of Field Theory and Some Mean-Field Features of Ising Models for Physical Review Letters, 47 (12), 886-886 DOI: 10.1103/PhysRevLett.47.886

Renata Jora (2015). $Φ^4$ theory is trivial arXiv arXiv: 1503.07298v1

Marco Frasca (2006). Proof of triviality of $λφ^4$ theory Int.J.Mod.Phys.A22:2433-2439,2007 arXiv: hep-th/0611276v5

Igor M. Suslov (2010). Asymptotic Behavior of the \Beta Function in the Φ^4 Theory: A Scheme
Without Complex Parameters J.Exp.Theor.Phys.111:450-465,2010 arXiv: 1010.4317v1

I. L. Bogolubsky, E. -M. Ilgenfritz, M. Müller-Preussker, & A. Sternbeck (2009). Lattice gluodynamics computation of Landau-gauge Green’s functions in the deep infrared Phys.Lett.B676:69-73,2009 arXiv: 0901.0736v3


Standard Model at the horizon

08/12/2014

ResearchBlogging.org

Hawking radiation is one of the most famous effects where quantum field theory combines successfully with general relativity. Since 1975 when Stephen Hawking uncovered it, this result has obtained a enormousStephen Hawking consideration and has been derived in a lot of different ways. The idea is that, very near the horizon of a black hole, a pair of particles can be produced one of which falls into the hole and the other escapes to infinity and is seen as emitted radiation. The overall effect is to drain energy from the hole, as the pair is formed at its expenses, and its ultimate fate is to evaporate. The distribution of this radiation is practically thermal and a temperature and an entropy can be attached to the black hole. The entropy is proportional to the area of the black hole computed at the horizon, as also postulated by Jacob Bekenstein, and so, it can only increase. Thermodynamics applies to black holes as well. Since then, the quest to understand the microscopic origin of such an entropy has seen a huge literature production with the notable Jacob Bekensteinunderstanding coming from string theory and loop quantum gravity.

In all the derivations of this effect people generally assumes that the particles are free and there are very good reasons to do so. In this way the theory is easier to manage and quantum field theory on curved spaces yields definite results. The wave equation is separable and exactly solvable (see here and here). For a scalar field, if you had a self-interaction term you are immediately in trouble. Notwithstanding this, in  the ’80 UnruhWilliam Unruh and Leahy, considering the simplified case of two dimensions and Schwarzschild geometry, uncovered a peculiar effect: At the horizon of the black the interaction appears to be switched off (see here). This means that the original derivation by Black-hole-model by Kip ThorneHawking for free particles has indeed a general meaning but, the worst conclusion, all particles become non interacting and massless at the horizon when one considers the Standard Model! Cooper will have very bad times crossing Gargantua’s horizon.

Turning back from science fiction to reality, this problem stood forgotten for all this time and nobody studied this fact too much. The reason is that the vacuum in a curved space-time is not trivial, as firstly noted by Hawking, and mostly so when particles interact. Simply, people has increasing difficulties to manage the theory that is already complicated in its simplest form. Algebraic quantum field theory provides a rigorous approach to this (e.g. see here). These authors consider an interacting theory with a \varphi^3 term but do perturbation theory (small self-interaction) probably hiding in this way the Unruh-Leahy effectValter Moretti.

The situation can change radically if one has exact solutions. A \varphi^4 classical theory can be indeed solved exactly and one can make it manageable (see here). A full quantum field theory can be developed in the strong self-interaction limit (see here) and so, Unruh-Leahy effect can be accounted for. I did so and then, I have got the same conclusion for the Kerr black hole (the one of Interstellar) in four dimensions (see here). This can have devastating implications for the Standard Model of particle physics. The reason is that, if Higgs field is switched off at the horizon, all the particles will lose their masses and electroweak symmetry will be recovered. Besides, further analysis will be necessary also for Yang-Mills fields and I suspect that also in this case the same conclusion has to hold. So, the Unruh-Leahy effect seems to be on the same footing and importance of the Hawking radiation. A deep understanding of it would be needed starting from quantum gravity. It is a holy grail, the switch-off of all couplings, kind of.

Further analysis is needed to get a confirmation of it. But now, I am somewhat more scared to cross a horizon.

V. B. Bezerra, H. S. Vieira, & André A. Costa (2013). The Klein-Gordon equation in the spacetime of a charged and rotating black hole Class. Quantum Grav. 31 (2014) 045003 arXiv: 1312.4823v1

H. S. Vieira, V. B. Bezerra, & C. R. Muniz (2014). Exact solutions of the Klein-Gordon equation in the Kerr-Newman background and Hawking radiation Annals of Physics 350 (2014) 14-28 arXiv: 1401.5397v4

Leahy, D., & Unruh, W. (1983). Effects of a λΦ4 interaction on black-hole evaporation in two dimensions Physical Review D, 28 (4), 694-702 DOI: 10.1103/PhysRevD.28.694

Giovanni Collini, Valter Moretti, & Nicola Pinamonti (2013). Tunnelling black-hole radiation with $φ^3$ self-interaction: one-loop computation for Rindler Killing horizons Lett. Math. Phys. 104 (2014) 217-232 arXiv: 1302.5253v4

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2014). Hawking radiation and interacting fields arXiv arXiv: 1412.1955v1


That’s a Higgs but how many?

17/11/2014

ResearchBlogging.org

CMS and ATLAS collaborations are yet up to work producing results from the datasets obtained in the first phase of activity of LHC. The restart is really near the corner and, maybe already the next summer, things can change considerably. Anyway what they get from the old data can be really promising and rather intriguing. This is the case for the recent paper by CMS (see here). The aim of this work is to see if a heavier state of Higgs particle exists and the kind of decay they study is Zh\rightarrow l^+l^-bb. That is, one has a signature with two leptons moving in opposite directions, arising from the dacy of the Z, and two bottom quarks arising from the decay of the Higgs particle. The analysis of this decay aims to get hints of existence of a heavier pseudoscalar Higgs state. This can be greatly important for SUSY extensions of the Standard Model that foresee more than one Higgs particle.

Often CMS presents its results with some intriguing open questions and also this is the case and so, it is worth this blog entry. Here is the main result

CMS study of Zh->llbbThe evidence, as said in the paper, is that there is a 2.6-2.9 sigma evidence at 560 GeV and a smaller one at around 300 GeV. Look elsewhere effect reduces the former at 1.1 sigma and the latter is practically negligible. Overall, this is pretty negligible but, as always, with more data at the restart, could become something real or just fade away. It should be appreciated the fact that a door is left open anyway and a possible effect is pointed out.

My personal interpretation is that such higher excitations do exist but their production rates are heavily suppressed with the respect to the observed ground state at 126 GeV and so, negligible with the present datasets. I am also convinced that the current understanding of the breaking of SUSY, currently adopted in MSSM-like to go beyond the Standard Model, is not the correct one provoking the early death of such models. I have explained this in a coupled of papers of mine (see here and here). It is my firm conviction that the restart will yield exciting results and we should be really happy to have such a powerful machine in our hands to grasp them.

Marco Frasca (2013). Scalar field theory in the strong self-interaction limit Eur. Phys. J. C (2014) 74:2929 arXiv: 1306.6530v5

Marco Frasca (2012). Classical solutions of a massless Wess-Zumino model J.Nonlin.Math.Phys. 20:4, 464-468 (2013) arXiv: 1212.1822v2


Living dangerously

05/11/2013

ResearchBlogging.org

Today, I read an interesting article on New York Times by Dennis Overbye (see here). Of course, for researchers, a discovery that does not open new puzzles is not really a discovery but just the end of the story. But the content of the article is intriguing and is related to the question of the stability of our universe. This matter was already discussed in blogs (e.g. see here) and is linked to a paper by Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, Gian F. Giudice, Gino Isidori, Alessandro Strumia (see here)  with the most famous picture

Stability and Higgs

Our universe, with its habitants, lives in that small square at the border between stability and meta-stability. So, it takes not too much to “live dangerously” as the authors say. Just a better measurement of the mass of the top quark can throw us there and this is in our reach at the restart of LHC. Anyhow, their estimation of the tunnel time is really reassuring as the required time is bigger than any reasonable cosmological age. Our universe, given the data coming from LHC, seems to live in a metastable state. This is further confirmed in a more recent paper by the same authors (see here). This means that the discovery of the Higgs boson with the given mass does not appear satisfactory from a theoretical standpoint and, besides the missing new physics, we are left with open questions that naturalness and supersymmetry would have properly assessed. The light mass of the Higgs boson, 125 GeV, in the framewrok of the Higgs mechanism, recently awarded with a richly deserved Nobel prize to Englert and Higgs, with an extensive use of weak perturbation theory is looking weary.

The question to be answered is: Is there any point in this logical chain where we can intervene to put all this matter on a proper track? Or is this the situation with the Standard Model to hold down to the Planck energy?

In all this matter there is a curious question that arises when you work with a conformal Standard Model. In this case, there is no mass term for the Higgs potential but rather, the potential gets modified by quantum corrections (Coleman-Weinberg mechanism) and a non-null vacuum expectation value comes out. But one has to grant that higher order quantum corrections cannot spoil conformal invariance. This happens if one uses dimensional regularization rather than other renormalization schemes. This grants that no quadratic correction arises and the Higgs boson is “natural”. This is a rather strange situation. Dimensional regularization works. It was invented by ‘t Hooft and Veltman and largely used by Wilson and others in their successful application of the renormalization group to phase transitions. So, why does it seem to behave differently (better!) in this situation? To decide we need a measurement of the Higgs potential that presently is out of discussion.

But there is a fundamental point that is more important than “naturalness” for which a hot debate is going on. With the pioneering work of Nambu and Goldstone we have learned a fundamental lesson: All the laws of physics are highly symmetric but nature enjoys a lot to hide all these symmetries. A lot of effort was required by very smart people to uncover them being very well hidden (do you remember the lesson from Lorentz invariance?). In the Standard Model there is a notable exception: Conformal invariance appears to be broken by hand by the Higgs potential. Why? Conformal invariance is really fundamental as all two-dimensional theories enjoy it. A typical conformal theory is string theory and we can build up all our supersymmetric models with such a property then broken down by whatever mechanism. Any conceivable more fundamental theory has conformal invariance and we would like this to be there also in the low-energy limit with a proper mechanism to break it. But not by hand.

Finally, we observe that all our theories seem to be really lucky: the coupling is always small and we can work out small perturbation theory. Also strong interactions, at high energies, become weakly interacting. In their papers, Gian Giudice et al. are able to show that the self-interaction of the Higgs potential is seen to decrease at higher energies and so, they satisfactorily apply perturbation theory. Indeed, they show that there will be an energy for which this coupling is zero and is due to change sign. As they work at high energies, the form of their potential just contains a quartic term. My question here is rather peculiar: What if exist exact solutions for finite (non-zero) quartic coupling that go like the inverse power of the coupling? We were not able to recover them with perturbation theory  but nature could have sat there. So, we would need to properly do perturbation theory around them to do the right physics. I have given some of there here and here but one cannot exclude that others exist. This also means that the mechanism of symmetry breaking can hide some surprises and the matter could not be completely settled. Never heard of breaking a symmetry by a zero mode?

So, maybe it is not our universe on the verge of showing a dangerous life but rather some of our views need a revision or a better understanding. Only then the next step will be easier to unveil. Let my bet on supersymmetry again.

Living Dangerously

Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miró, José R. Espinosa, Gian F. Giudice, Gino Isidori, & Alessandro Strumia (2012). Higgs mass and vacuum stability in the Standard Model at NNLO JHEP August 2012, 2012:98 arXiv: 1205.6497v2

Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, Gian F. Giudice, Filippo Sala, Alberto Salvio, & Alessandro Strumia (2013). Investigating the near-criticality of the Higgs boson arXiv arXiv: 1307.3536v1

Marco Frasca (2009). Exact solutions of classical scalar field equations J.Nonlin.Math.Phys.18:291-297,2011 arXiv: 0907.4053v2

Marco Frasca (2013). Exact solutions and zero modes in scalar field theory arXiv arXiv: 1310.6630v1


Much closer to the Standard Model

18/03/2013

ResearchBlogging.org

Today, the daily from arxiv yields a contribution from John Ellis and Tevong You analyzing new data presented at Aspen and Moriond the last two weeks by CMS and ATLAS about Higgs particle (see here). Their result can be summarized in the following figure

Ellis & You: agreement with Standard Modelthat is really impressive. This means that the updated data coming out from LHC constraints even more the Higgs particle found so far to be the Standard Model one. Another impressive conclusion they are able to draw is that the couplings appear to be proportional to the masses as it should be expected from a well-behaved Higgs particle. But they emphasize that this is “a” Higgs particle and the scenario is well consistent with supersymmetry. Citing them:

The data now impose severe constraints on composite alternatives to the elementary Higgs boson of the Standard Model. However, they do not yet challenge the predictions of supersymmetric models, which typically make predictions much closer to the Standard Model values. We therefore infer that the Higgs coupling measurements, as well as its mass, provide circumstantial support to supersymmetry as opposed to these minimal composite alternatives, though this inference is not conclusive.

They say that further progress on the understanding of this particle could be granted after the upgraded LHC will run and, indeed, nobody is expecting some dramatic change into this scenario from the data at hand.

John Ellis, & Tevong You (2013). Updated Global Analysis of Higgs Couplings arXiv arXiv: 1303.3879v1


Unbreakable

14/11/2012

Today, at HCP2012, new results on Higgs boson search were made available by CMS and ATLAS. Of course, well aligned with preceding rumors, all in all these appear rather disappointing. Maybe, beyond the increasing agreement with Standard Model expectations, the most delusional result is that the particle announced on July 4th appears to be completely lonely on a desert ranging till almost 1 TeV, at least if one is looking for other Higgs particles behaving Standard Model-like. \tau\tau decay rate is now aligning with expectations even if there is some room for a different outcome. On the other side, both experiments did not update \gamma\gamma findings. The scenario that is emerging from these results is the theorist’s nightmare. Tomorrow, all this will be collected in single talks by CMS and ATLAS speakers.

In a retrospective we could say that people claiming for a prize to discoverers of the Higgs mechanism seem to be vindicated. There appears no sign of supersymmetry that is more and more relinquished in a nowhere land. But, I would like to point out that, if a supersymmetric theory is the right one, there is just one theory to be singled out exploring the parameter space. It is normal in any case to see such a vast epidemic death of theories. It is also possible that theorists should now do a significant effort for new proposals beyond those largely explored in these last thirty years.

Standard Model is even more resembling a perfect theory really unbreakable and mimicking the success of the Maxwell equations put forward 150 years ago.  But we know it must break…

Finally, I would like to conclude this rather fizzling out post by pointing out a rather funny side of this situation. Tommaso Dorigo has a bet on with Gordon Watts and Jacques Distler amounting to $1200 on the non-existence of SUSY partners. This bet has not been payed yet as Distler is claiming there are a lot of “juicy rumors” from CERN and the terms are not fulfilled yet (see comments here).  I do not know what rumors Distler is talking about but, unless CERN is not hiding data (that would appear a rather strange behavior at best), maybe it is time to do a check on who the winner could be.


Mass generation: The solution

26/12/2010

ResearchBlogging.org

In my preceding post I have pointed out an interesting mathematicalquestion about the exact solutions of the scalar field theory that I use in this paper

\Box\phi+\lambda\phi^3=0

given by

\phi=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x,i)

that holds for

p^2=\mu^2\sqrt{\frac{\lambda}{2}}.

If you compute the Hamiltonian the energy does not appear to be finite, differently from what the relation dispersion is saying. This is very similar to what happens to plane waves for the wave equation. The way out is to take a finite volume and normalize properly the plane waves. One does this to get the integral of the Hamiltonian finite and all amounts to a proper normalization. In our case where must this normalization enter? The striking answer is: The coupling. This is an arbitrary parameter of the theory and we can properly rescale it to get the right normalization in the Hamiltonian. The final result is a running coupling exactly in the same way as I and others have obtained for the quantum theory. You can see the coupling entering in the right way both in the solution and in the computation of the Hamiltonian.

If you are curious about these computations you can read the revised version of my paper to appear soon on arxiv.

Marco Frasca (2010). Mass generation and supersymmetry arxiv arXiv: 1007.5275v1


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